Complete group

As an example, all the symmetric groups, , are complete except when {{math|n ∈ {2, 6}}}. For the case , the group has a non-trivial center, while for the case , there is an outer automorphism. The automorphism group of a simple group is an almost simple group; for a non-abelian simple group , the automorphism group of is complete. == Properties ==

A complete group is always isomorphic to its automorphism group (via sending an element to conjugation by that element), although the converse need not hold: for example, the dihedral group of 8 elements is isomorphic to its automorphism group, but it is not complete. For a discussion, see . == Extensions of complete groups ==

Assume that a group is a group extension given as a short exact sequence of groups : with kernel, , and quotient, . If the kernel, , is a complete group then the extension splits: is isomorphic to the direct product, . A proof using homomorphisms and exact sequences can be given in a natural way: The action of (by conjugation) on the normal subgroup, , gives rise to a group homomorphism, . Since and has trivial center the homomorphism is surjective and has an obvious section given by the inclusion of in . The kernel of is the centralizer of in , and so is at least a semidirect product, , but the action of on is trivial, and so the product is direct. This can be restated in terms of elements and internal conditions: If is a normal, complete subgroup of a group , then is a direct product. The proof follows directly from the definition: is centerless giving is trivial. If is an element of then it induces an automorphism of by conjugation, but and this conjugation must be equal to conjugation by some element of . Then conjugation by is the identity on and so is in and every element, , of is a product in . == References ==